Generalized Lanczos Algorithm for Variational Quantum Monte Carlo

Abstract

We show that the standard Lanczos algorithm can be efficiently implemented statistically and self consistently improved, using the stochastic reconfigurat ion method, which has been recently introduced to stabilize the Monte Carlo sign problem instability. With this scheme a few Lanczos steps over a given variational wavefunction are possible even for large size as a particular case of a more general and more accurate technique that allows to obtain lower variational energies. This method has been tested extensively for a strongly correlated model like the t-J model. With the standard Lanczos technique it is possible to compute any kind of correlation functions, with no particular computational effort. By using that the variance $<H^2>-<H>^2$ is zero for an exact eigenstate, we show that the approach to the exact solution with few Lanczos ite rations is indeed possible even for $\sim 100$ electrons for reasonably good initial wavefunctions.The variational stochastic reconfiguration technique presented here allows in general a many-parameter energy optimization of any computable many-body wavefunction, including for inst ance generic long range Jastrow factors and arbitrary site dependent orbital determinants. This scheme improves further the accuracy of the calculation, especially for long distance correlation functions.